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1 Principles and Lecture 2 of 4-part series Capital Spring School on Risk, Insurance and Finance European University at St. Petersburg, Russia 2-4 April 2012 Fair Wang s University of Connecticut, USA page 1

2 Outline 1 Capital Fair Wang s 2 Capital 3 Fair 4 Wang s 5 page 2

3 Capital Fair Wang s Alternative definitions of economic The Specialty Guide on Economic Capital, page 7, prepared by the SOA Risk Task Force (RMTF) headed by Hubert Mueller in 2003, offers the following alternative definitions of economic : Definition #1 Economic is defined as sufficient surplus to meet potential negative cash flows and reductions in value of assets or increases in value of liabilities at a given level of risk tolerance, over a specified time horizon. Definition #2 Economic is defined as the excess of the market value of the assets over the fair value of liabilities required to ensure that obligations can be satisfied at a given level of risk tolerance, over a specified time horizon. Definition #3 Economic is defined as sufficient surplus to maintain solvency at a given level of risk tolerance, over a specified time horizon. page 3

4 Capital Fair Wang s The purpose of economic Calculating the economic for a firm has its many purposes (this list not all encompassing): understand the company s overall risk profile budgeting pricing purposes company asset and liability management understanding company s risk tolerances and possible constraints performance measurement and incentive compensation regulatory and other external parties (e.g. investors, ratings agencies) Many of these can be found in the Specialty Guide; a lot of development since its publication. page 4

5 Capital and product pricing For simplicity, consider an insurer faced with pricing a risk Z, only for a given short horizon, and assume the company has risk tolerance described by an exponential utility function: Capital Fair Wang s u(x) = 1 a( 1 e ax ), for a > 0. It is rather straightforward to show that the smallest amount of premium this company is willing to accept in exchange of facing the risk can be expressed as: P = 1 a log E[ e az ] Now, suppose the company is faced with the task of pricing a portfolio of n of these identical risks, say X 1, X 2,..., X n so that the aggregate risk is S = X 1 + X X n page 5

7 The problem Capital Fair Wang s Consider a portfolio of n individual losses X 1,..., X n during some well-defined reference period. Assume these random losses have a dependency structure characterized by the joint distribution of the random vector (X 1,..., X n ). The aggregate loss is the sum S = n i=1 X i, or in some instances, it could be a weighted sum S = n i=1 w ix i. Assume company holds aggregate level of K which may be determined from a risk measure ρ such that K = ρ(s) R. Here the (economic) is the smallest amount the company must set aside to withstand aggregated losses at an acceptable level, and tolerance. page 7

8 - continued Capital Fair Wang s The company now wishes to allocate K across its various business units. determine non-negative real numbers K 1,..., K n satisfying: n K i = K. This requirement is referred to as the full requirement. We will see that this requirement is a constraint in our optimization problem (in the optimal paper). i=1 page 8

11 Some notation Denote by X T = (X 1, X 2,..., X n ) the vector of losses, with each entry denoting the loss for the applicable line of business. Define an A to be a mapping A : X T R n such that Capital Fair Wang s A ( X T ) = ( K 1, K 2,..., K n ), where the full is satisfied: K = ρ[z ] = Each component K i of the is viewed as the i-th line of business contribution to the total company. Because must also reflect the fact that each line operates in the presence of the other lines, the notation K i = A ( X i X 1,..., X n ) n i=1 K i may be well suited for this purpose. page 11

13 Capital Fair Wang s page 13 Intuitive interpretations The no undercut criterion: recognizes the benefits of diversification risk allocated to a business unit may not exceed the risk allocated if these were offered on a stand-alone basis analogous to sub-additivity for a coherent risk measure The symmetry criterion: states that two lines of business that equally contribute to the risk within the firm must have equal must therefore recognize only the level or degree of contribution to risk within the firm, and nothing else sometimes called equitability property The consistency criterion: ensures that a unit s cannot depend on the level at which the occurs is independent of the hierarchical structure of the firm

14 Relative or proportional method Capital Fair Although quite a popular method, partly because of its simplicity, the relative formula according to A ( X i X 1,..., X n ) = ρ[x i ] ρ[x 1 ] + + ρ[x n ] ρ[s] violates all three possible criteria for a fair. See Valdez and Chernih (2003) for proofs. Wang s page 14

16 - continued Capital Fair Wang s The formula based on the covariance principle satisfies the three criterion or properties of a possible fair : no undercut, symmetry, and consistency. See Valdez and Chernih (2003) for proofs. page 16

18 Capital Fair Wang s - continued For the aggregate loss S, its risk premium is given by [ n ] ρ[s] = ρ X i = H λ [S, S] E[S]. i=1 It is rather straightforward to show that ρ[x i ] = Cov[X i, exp(λs] E[exp(λS)] and ρ[s] = Cov[S, exp(λs]. E[exp(λS)] Wang (2002) proposed computing the of to individual business unit i based on the following formula: K i = H λ [X i, S] E[X i ]. Assuming an aggregate of K, the parameter λ can be computed by solving then K = H λ [S, S] E[S]. page 18 It is not difficult to show that the full requirement is met in this case: K = n i=1 K i.

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